Hermitian-symmetric matrix operator

Note that on a finite-dimensional vector space V, a linear operator is Her-mitian if and only if T = T. More concretely, in C", a linear operator is Hermitian-symmetric if and only if its matrix M in the standard basis satis-lies M = M, where M denotes the conjugate transpose matrix. To check that a hnear operator is Hermitian, it suffices to check Equation 3.2 on basis vectors. Physics textbooks often contain expressions such as (+z H — z). These expressions are well defined only if H is a Hermitian operator. If H yNQK not Hermitian, the value of the expression would depend on where one applies the H. 

The diagonal elements of a Hermitian matrix must be real. A real symmetric matrix is a special case of a Hermitian matrix. (The relation between Hermitian matrices and Hermitian operators will be shown in Section 2.3.)... 

Bap being a hermitian matrix, and Aap a symmetric matrix. According to the Tyablikov-Bogoljubov method66 the diagonalization of the quadratic form (A.l) can be performed by introducing new operators... 

The coefficient at A describes the linear response of the quantity A to the perturbation W. It can be given a rather more symmetric form. Indeed the amplitude of the j-th unperturbed state in the correction to the fc-th state is proportional to some skew Hermitian operator (the perturbation matrix W is Hermitian, but the denominator changes its sign when the order of the subscripts changes). With this notion and assuming that Wkk = 0 (see above) we can remove the restriction in the summation and write ... 

Some types of matrices that appear frequently are Hermitian matrices, for which A = A anti-Hermitian matrices, for which A = — A and unitary matrices, for which = U The MCSCF methods diseussed in most detail in this review will involve only operations of real matrices. In this case these matrix types reduce to symmetric matrices, for which A = A , antisymmetric matrices, for which A = — A, and orthogonal matrices, for which U = U. A particular type of orthogonal matrix is called a rotation matrix and satisfies the relation Det(R)= -I-1, where Det(R) is the usual definition of a determi-... 

To one who is familiar with nonrelativistic quantum mechanic it may appear quite clear what is meant by a spherically symmetric potential— any potential that actually only depends on x, so that it is invariant under any rotation applied to the system. It is indeed true that a scalar function

R ) for all rotation matrices R, if and only if it is a function of r = [x]. But a general potential in quantum mechanics is given by a Hermitian matrix, and the unitary operators (86) representing the rotations in the Hilbert space of the Dirac equation can also affect the spinor-components. Hence it is not quite straightforward to tell, which potentials are spherically symmetric. 

Here the W are operators of the subsystem and the superscript dagger denote the Hermitian conjugate. The Redfield equation can be written in this form only when an additional symmetrization of the bath correlation functions is performed [48]. Note that this alternative equation also expresses the dissipative evolution of the density matrix in terms of N x N... 

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